Jacobi coordinates

File:Two-body Jacobi coordinates.JPG; Jacobi coordinates are $\backslash boldsymbol\{R\}=\backslash frac\; \{m\_1\}\{M\}\; \backslash boldsymbol\{x\}\_1\; +\; \backslash frac\; \{m\_2\}\{M\}\; \backslash boldsymbol\{x\}\_2$ and $\backslash boldsymbol\{r\}\; =\; \backslash boldsymbol\{x\}\_1\; -\; \backslash boldsymbol\{x\}\_2$ with $M\; =\; m\_1+m\_2$.{{cite book |title=Differential Equations |author=David Betounes |url=https://archive.org/details/differentialequa0000beto |url-access=registration |isbn=0-387-95140-7 |page=58; Figure 2.15 |date=2001 |publisher=Springer}}]]

File:Jacobi coordinates — illustration for four bodies.svg

In the theory of many-particle systems, Jacobi coordinates often are used to simplify the mathematical formulation. These coordinates are particularly common in treating polyatomic molecules and chemical reactions,

{{cite book |title=Theory and application of quantum molecular dynamics |author= John Z. H. Zhang |url=https://books.google.com/books?id=b8AzpUPopqQC&pg=PA104 |page=104 |isbn=981-02-3388-4 |date=1999 |publisher=World Scientific}}

and in celestial mechanics.

For example, see {{cite book |title=Capture Dynamics and Chaotic Motions in Celestial Mechanics |author= Edward Belbruno |url=https://books.google.com/books?id=dK-fl0KrOEIC&pg=PA9 |page=9 |isbn=0-691-09480-2 |date=2004 |publisher=Princeton University Press}}

An algorithm for generating the Jacobi coordinates for N bodies may be based upon binary trees.

{{cite book |title=Classical and celestial mechanics |author= Hildeberto Cabral, Florin Diacu |chapter-url=https://books.google.com/books?id=q1emz4C4lYQC&pg=PA230 |page=230 |chapter=Appendix A: Canonical transformations to Jacobi coordinates |isbn=0-691-05022-8 |publisher=Princeton University Press |date=2002}}

In words, the algorithm may be described as follows:

We choose two of the N bodies with position coordinates x_j and x_k and we replace them with one virtual body at their centre of mass. We define the relative position coordinate r_jk = x_j − x_k.

We then repeat the process with the N − 1 bodies consisting of the other N − 2 plus the new virtual body. After N − 1 such steps we will have Jacobi coordinates consisting of the relative positions and one coordinate giving the position of the last defined centre of mass.

For the N-body problem the result is:{{cite book |title=Advanced electromagnetism and vacuum physics |author=Patrick Cornille |page=102 |chapter-url=https://books.google.com/books?id=y8sSFTDkQ20C&pg=PA102 |chapter=Partition of forces using Jacobi coordinates |isbn=981-238-367-0 |date=2003 |publisher=World Scientific}}

:$\backslash boldsymbol\{r\}\_j=\; \backslash frac\{1\}\{m\_\{0j\}\}\; \backslash sum\_\{k=1\}^j\; m\_k\backslash boldsymbol\; \{x\}\_k\; \backslash \; -\; \backslash \; \backslash boldsymbol\{x\}\_\{j+1\}\backslash \; ,\; \backslash quad\; j\; \backslash in\; \backslash \{1,\; 2,\; \backslash dots,\; N-1\backslash \}$

:$\backslash boldsymbol\{r\}\_N=\; \backslash frac\{1\}\{m\_\{0N\}\}\; \backslash sum\_\{k=1\}^N\; m\_k\backslash boldsymbol\; \{x\}\_k\; \backslash \; ,$

with

:$m\_\{0j\}\; =\; \backslash sum\_\{k=1\}^j\; \backslash \; m\_k\; \backslash \; .$

The vector $\backslash boldsymbol\{r\}\_N$ is the center of mass of all the bodies and $\backslash boldsymbol\{r\}\_1$ is the relative coordinate between the particles 1 and 2:

The result one is left with is thus a system of N-1 translationally invariant coordinates $\backslash boldsymbol\{r\}\_1,\; \backslash dots,\; \backslash boldsymbol\{r\}\_\{N-1\}$ and a center of mass coordinate $\backslash boldsymbol\{r\}\_N$, from iteratively reducing two-body systems within the many-body system.

This change of coordinates has associated Jacobian equal to $1$.

If one is interested in evaluating a free energy operator in these coordinates, one obtains

:$H\_0=-\backslash sum\_\{j=1\}^N\backslash frac\{\backslash hbar^2\}\{2\; m\_j\}\backslash ,\; \backslash nabla^2\_\{\backslash boldsymbol\{x\}\_j\}\; =\; -\backslash frac\{\backslash hbar^2\}\{2\; m\_\{0N\}\}\backslash ,\backslash nabla^2\_\{\backslash boldsymbol\{r\}\_\{N\}\}\backslash !-\backslash frac\{\backslash hbar^2\}\{2\}\backslash sum\_\{j=1\}^\{N-1\}\backslash !\backslash left(\backslash frac\{1\}\{m\_\{j+1\}\}+\backslash frac\{1\}\{m\_\{0j\}\}\backslash right)\backslash nabla^2\_\{\backslash boldsymbol\{r\}\_j\}$

In the calculations can be useful the following identity

:$\backslash sum\_\{k=j+1\}^N\; \backslash frac\{m\_k\}\{m\_\{0k\}m\_\{0k-1\}\}=\backslash frac\{1\}\{m\_\{0j\}\}-\backslash frac\{1\}\{m\_\{0N\}\}$.